Möbius strip
As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE.The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-track roller coasters whose carriages alternate between the two tracks, and world maps printed so that antipodes appear opposite each other.Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in social choice theory.In popular culture, Möbius strips appear in artworks by M. C. Escher, Max Bill, and others, and in the design of the recycling symbol.Performers including Harry Blackstone Sr. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip.[3] Another mosaic from the town of Sentinum (depicted) shows the zodiac, held by the god Aion, as a band with only a single twist.There is no clear evidence that the one-sidedness of this visual representation of celestial time was intentional; it could have been chosen merely as a way to make all of the signs of the zodiac appear on the visible side of the strip.[4] Another use of this surface was made by seamstresses in Paris (at an unspecified date): they initiated novices by requiring them to stitch a Möbius strip as a collar onto a garment.It is a non-orientable surface: if an asymmetric two-dimensional object slides one time around the strip, it returns to its starting position as its mirror image.[6] A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a chiral object with right- or left-handedness.[12] Möbius strips with odd numbers of half-twists greater than one, or that are knotted before gluing, are distinct as embedded subsets of three-dimensional space, even though they are all equivalent as two-dimensional topological surfaces.[13] More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine the same knot and they have the same number of twists as each other.This transformation is an example of a deformation retraction, and its existence means that the Möbius strip has many of the same properties as its centerline, which is topologically a circle.[6] These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called paradromic rings.[28][29] The Möbius strip can also be embedded as a polyhedral surface in space or flat-folded in the plane, with only five triangular faces sharing five vertices.[37] A rectangular Möbius strip, made by attaching the ends of a paper rectangle, can be embedded smoothly into three-dimensional space whenever its aspect ratio is greater than[44][45] If the requirement of smoothness is relaxed to allow continuously differentiable surfaces, the Nash–Kuiper theorem implies that any two opposite edges of any rectangle can be glued to form an embedded Möbius strip, no matter how small the aspect ratio becomes.[52] Lawson's Klein bottle is a self-crossing minimal surface in the unit hypersphere of 4-dimensional space, the set of points of the form[95][96] It is also a popular subject of mathematical sculpture, including works by Max Bill (Endless Ribbon, 1953), José de Rivera (Infinity, 1967), and Sebastián.[100] Some variations of the recycling symbol use a different embedding with three half-twists instead of one,[99] and the original version of the Google Drive logo used a flat-folded three-twist Möbius strip, as have other similar designs.[102] The Möbius strip has also featured in the artwork for postage stamps from countries including Brazil, Belgium, the Netherlands, and Switzerland.However, many of these are projects or conceptual designs rather than constructed objects, or stretch their interpretation of the Möbius strip beyond its recognizability as a mathematical form or a functional part of the architecture.[105][106] An example is the National Library of Kazakhstan, for which a building was planned in the shape of a thickened Möbius strip but refinished with a different design after the original architects pulled out of the project.[107] One notable building incorporating a Möbius strip is the NASCAR Hall of Fame, which is surrounded by a large twisted ribbon of stainless steel acting as a façade and canopy, and evoking the curved shapes of racing tracks.[108] On a smaller scale, Moebius Chair (2006) by Pedro Reyes is a courting bench whose base and sides have the form of a Möbius strip.[109] As a form of mathematics and fiber arts, scarves have been knit into Möbius strips since the work of Elizabeth Zimmermann in the early 1980s.[110] In food styling, Möbius strips have been used for slicing bagels,[111] making loops out of bacon,[112] and creating new shapes for pasta.[113] Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked in speculative fiction as the basis for a time loop into which unwary victims may become trapped.Examples of this trope include Martin Gardner's "No-Sided Professor" (1946), Armin Joseph Deutsch's "A Subway Named Mobius" (1950) and the film Moebius (1996) based on it.[114] Other works of fiction have been analyzed as having a Möbius strip–like structure, in which elements of the plot repeat with a twist; these include Marcel Proust's In Search of Lost Time (1913–1927), Luigi Pirandello's Six Characters in Search of an Author (1921), Frank Capra's It's a Wonderful Life (1946), John Barth's Lost in the Funhouse (1968), Samuel R. Delany's Dhalgren (1975) and the film Donnie Darko (2001).
mathematicssurfaceJohann Benedict ListingAugust Ferdinand Möbiusnon-orientableclockwisetopological spaceEuclidean spaceknottedtopologically equivalentboundary curveruled surfacedevelopable surfacetrihexaflexagonminimal surfacehyperspheresurfaces of constant curvaturemechanical beltsroller coastersworld mapsantipodessocial choice theoryM. C. EscherMax Billrecycling symbolNASCAR Hall of FameHarry Blackstone Sr.Thomas Nelson DownscanonsJ. S. Bachspeculative fictionSentinumChain pumpIsmail al-Jazariuntwisted ringszodiacourobourosfigure-eightnon-orientable surfaceorientable surfacesCartesian productuncountable setdisjointchiralannulusdeformation retractionfundamental groupinfinite cyclic grouphomotopyTietze's graphthree utilities problemfour color theoremRingel–Youngs theoremdual graphcomplete graphdrawn without crossings on a planeembeddedMöbius ladderscomplete bipartite graphEuler characteristicPlücker's conoidparametric surfaceCartesian coordinatessolid torusequilateral triangleaspect ratiocross sectionpolyhedral surfacecylindersimplicial complexfour-dimensional regular simplexhyperplanesquadrilateralsoctahedronprojective plane(more unsolved problems in mathematics)Richard Schwartzcontinuously differentiableNash–Kuiper theoremtautological line bundleplate theoryMichael Sadowskyalgebraic surfacesboundarycircleunknottedKlein bottleimmersedunit hyperspheregreat circleStereographic projectionorthogonal grouplevel setsquadrilateralpinch pointWhitney umbrellarelative interiorRiemannian geometryGaussian curvaturegeodesicsquotient spaceglide reflectionflat manifoldsupper half plane (Poincaré) modelhyperbolic planehalf-planeonce-puncturedminimal surfacesmean curvatureWilliam Hamilton Meeks, IIIBjörling problemreal projective planeline at infinityprojective dualityunordered pairsaffine transformationsMöbius transformationsLie groupsalgebraic structurehomogeneous spacesolvmanifoldscounterexamplenilmanifolddirect productcompactstabilizer subgroup